ps2fpe - r) and Y(s) in Figure 3.55 Block diagram for...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: r) and Y(s) in Figure 3.55 Block diagram for Problem 3.22 Figure 3.56 Circuit for Problem 3.23 Figure 3.57 Unity feedback system for Problem 3.24 Problems 159 Problems for Section 3.3: Effect of Pole Locations 3.23 For the electric circuit shown in Fig. 3.56, find the following: (a) The time—domain equation relating [(r) and 121(1); (b) The lime-domain equation relating i(t) and v20); Vol (c) Assuming all initial conditions are zero, the transfer function 1;) and the damping ratio 4 and undamped natural frequency w" of the system; ' (d) The values of R that will result in v20) having an overshoot of no more than 25%, assuming v10) is a unit step, L = 10 mH. and C = 4 #F. L R mm It!) C u-p(l) _ i L 3.24 For the unity feedback system shown in Fig. 3.57. specify the gain K of the proportional controller so that the output y(t) has an overshoot of no more than 10% in response to a unit step. 3.25 For the unity feedback system shown in Fig. 3.58, specify the gain and pole location of the compensator so that the overall closed-loop response to a unit-step input has an overshoot of no more than 25%. and a 1% settling time of no more than 0.1 sec. Verify your design using MATLAB. 160 Chapter3 Dynamic Response Figure 3.58 Unity feedback system for Problem 3.25 Figure 3.59 Unity feedback system for Problem 3.28 Figure 3.60 Desired closed-loop pole locations for Problem 3.28 Problemsfor Section 3.4: Time-Domain Specification 3.26 3.27 3.28 Suppose you desire the peak time of a given second-order system to be less than 1”, Draw the region in the s-plane that corresponds to values of the poles that meet the specification (p < 4,. A certain servomechanism system has dynamics dominated by a pair of complex poles and no finite zeros. The time-domain specifications on the rise time (t,-), percent overshoot (Mp), and settling time ([5) are given by t, 5 0.6 sec , Mp 5 l7%, Is 5 9.2 sec. (a) Sketch the region in the s-plane where the poles could be placed so that the system will meet all three specifications. (b) Indicate on your sketch the specific locations (denoted by x) that will have the smallest rise-time and also meet the settling time specification exactly. Suppose you are to design a unity feedback controller for a first-order plant depicted in Fig. 3.59. (As you will learn in Chapter 4, the configuration shown is referred to as a proportional-integral controller.) You are to design the controller so that the closed-loop poles lie within the shaded regions shown in Fig. 3.60. (a) What values of w" and 1; correspond to the shaded regions in Fig. 3.59? (A simple estimate from the figure is sufficient.) (b) Let Ka‘ = a = 2. Find values for K and K, so that the poles of the closed-loop system lie within the shaded regions Problems 161 (c) Pr0ve that no matter what the values of K0 and or are, the controller provides enough flexibility to place the poles anywhere in the complex (left—halt) plane. 3.29 The open-loop transfer function of a unity feedback system is K s(s+2)' 0(3) :- The desired system response to a step input is specified as peak time t,, = 1 sec and overshoot M,, = 5%. (:1) Determine whether both specifications can be met simultaneously by selecting the right value of K. (b) Sketch the associated region in the s-plane where both Specifications are met, and indicate what root locations are possible for some likely values of K. (c) Relax the specifications in part (a) by the same factor and pick a suitable value for K . and use MATLAB to verify that the new specifications are satisfied. 3.30 The equations of motion for the DC motor shown in Fig. 2.32 were given in Eqs. (2.52— 2.53) as .. K K . K JQO ‘i' (b + 0m : [T1471- ll (1 Assume that J,,, = 0.01 kg-mz. b = 0.001 N-m-scc, Kc = 0.02 V-sec, K, = 0.02 N-m/A, a = 10 S2. (3) Find the transfer function between the applied voltage Va and the motor speed 9,". (b) What is the steady—state speed of the. motor after a voltage V(( = 10 V has been applied? . (c) Find the transfer function between the applied voltage v“ and the shaft angle 6,". (d) Suppose feedback is added to the system in part (c) so that it. becomes a position servo device such that the applied voltage is given by "a = Kigr _ am) Where K is the feedback gain. Find the transfer function between 6,. and 9m. (9) What is the maximum value of K that can be used if an overshoot Mp < 20% is desired? (f) What values of K will provide a rise time of less than 4 sec? (Ignore the Mp constraint.) (g) Use MATLAB to plot the step response of the position sen/o system for values of the gain K = 0.5, 1, and 2. Find the overshoot and rise time for each of the three step responses by examining your plots. Are the plots consistent with your calculations in parts (e) and (f)? 3.31 You wish to control the elevation of the satellite-tracking antenna shown in Figs. 3.61 and 3.62. The antenna and drive parts have a moment of inertia J and a damping B; 162 Chapter 3 Dynamic Response Figure 3.61 Satellite—tracking antenna Source: Courtesy Space Systems/Lora! Figure 3.62 Schematic ofantenna for Problem 3.31 these arise to some extent from bearing and aerodynamic friction, but mostly from the back emf of the DC drive motor. The equations of motion are ’I’ = To where Tr is the torque from the drive motor. Assume that J = 600.000 kg-m3 B = 20,000 N-msec. (3) Find the transfer function between the applied torque '1}; and the antenna angle 6, (b) Suppose the applied torque is computed so that 9 tracks a referenCe command 0, according to the feedback law Tc = Kwr — 0), where K is the feedback gain. Find the transfer function between 0, and 9. Problems 163 (c) What is the maximum value of K that can be used ifyou wish to have an overshoot M,, < 10%? (d) What values of K will provide a rise time of less than 80 sec? (Ignore the Mp constraint.) A (9) Use MATLAB to plot the step response of the antenna system for K = 200. 400, l000, and 2000. Find the overshoot and rise time of the four step responses by examining your plots. Do the plots confirm your calculations in parts (c) and (d)? 3.32 Show that the second-order system 5- + 25%;, + wfiy = 0. y(0) = yo. 540) = 0. has the response 8-0! 1 NU) = )‘o— sin(wd! + cos— t/l — (2 Prove that, for the underdamped case (C < l). the response oscillations decay at a predictable rate (see Fig. 3.63) called the logarithmic decrement C). Yo 2”; 5=ln'— :lne‘n" =ard : —— )‘I \/l — g2 A' A = ln ii 2 ln l. 3'] )‘i where 271 2n 1'11: — = wd (um/l — r2 is the damped natural period of vibration. The damping coefficient in terms of the logarithmic decrement is then _ 5 \/4n2 +52' C Figure 3.63 Definition of logarithmic decrement ...
View Full Document

This document was uploaded on 02/22/2012.

Page1 / 5

ps2fpe - r) and Y(s) in Figure 3.55 Block diagram for...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online